This lecture is intended for those who are not acquainted with the theory of computational complexity. For this reason, I shall talk only about the foundations of this theory and its very first results. I shall try to deliver the main ideas which guide the researchers in this field of science.

The setting of my narrative are stories about one personage called M (for “mathematician”). I shall start with the following story.

Prehistory

Once upon a time, M sat at home trying to prove some (maybe important, or maybe not) theorem T. He tried to prove this theorem for a week, two weeks, a month, …, but with no result. In the end, he gave up and asked quite the natural question:

The question was addressed to nobody; most likely, it would fly away if another personage, L (for “logician”), would not pass by.

L heard the question, came into the room, and explained that such questions started to interest mathematicians some time in the beginning of the twentieth century. In a general setting, this question is contained in the famous Hilbert's Program devoted to the notion of mathematical proof.

This program, in particular, included the following three items.

1. Formalization of the notion of proof. Before asking whether or not an assertion can be proved, we must give a rigorous mathematical definition of provability.

2. […]